Why does the Botany Bay have to have artificial gravity? The decks could be aligned such that they are aimed fore and aft rather than along the centerline (as on the Enterprise). As a result, the motion ofmthe craft in a forward direction at a steady rate would allow for the sensation of gravity without the use of any artificial gravity machinery.

Steady motion doesn't generate the feel of gravity - only acceleration motion does. And the engines of this relic weren't working...

The Phoenix did not seem to have inertial damping or and AGS.

Well, the rocket propelled our heroes to high Earth-escape orbit at such speed that lack of inertial damping would probably have been very noticeable - the trio would have fainted from the acceleration!

Probably the artificial gravity switched on then, too.

The problem with that is that the sleeping chambers didn't have safety belts. Half the supermen would have ended up face down when the gravity slammed back on!

As to why the Botany Bay had some technological advancements beyond what the Phoenix had, first of all the Phoenix was built under much more pitiful conditions. But moreover, it was only a stripped down testbed for the warp drive, and not any sort of craft intended for extended flight.